Boundary layer approach in the modeling of breaking solitary wave runup

The boundary layer is very important in the relation between wave motion and bed stress, such as sediment transport. It is a known fact that bed stress behavior is highly influenced by the boundary layer beneath the waves. Specifically, the boundary layer underneath wave runup is difficult to assess and thus, it has not yet been widely discussed, although its importance is significant. In this study, the shallow water equation (SWE) prediction of wave motion is improved by being coupled with the k–ω model, as opposed to the conventional empirical method, to approximate bed stress. Subsequently, the First Order Center Scheme and Monotonic Upstream Scheme of Conservation Laws (FORCE MUSCL), which is a finite volume shock-capturing scheme, is applied to extend the SWE range for breaking wave simulation. The proposed simultaneous coupling method (SCM) assumes the depth-averaged velocity from the SWE is equivalent to free stream velocity. In turn, free stream velocity is used to calculate a pressure gradient, which is then used by the k–ω model to approximate bed stress. Finally, this approximation is applied to the momentum equation in the SWE. Two experimental cases will be used to verify the SCM by comparing runup height, surface fluctuation, bed stress, and turbulent intensity values. The SCM shows good comparison to experimental data for all before-mentioned parameters. Further analysis shows that the wave Reynolds number increases as the wave propagates and that the turbulence behavior in the boundary layer gradually changes, such as the increase of turbulent intensity.     

Run-up of solitary waves on slopes with different profiles

The problem of sea-wave run-up on a beach is discussed within the framework of exact solutions of a nonlinear theory of shallow water. Previously, the run-up of solitary waves with different forms (Gaussian and Lorentzian pulses, a soliton, special-form pulses) has already been considered in the literature within the framework of the same theory. Depending on the form of the incident wave, different formulas were obtained for the height of wave run-up on a beach. A new point of this study is the proof of the universality of the formula for the maximum height of run-up of a solitary wave on a beach for the corresponding physical choice of the determining parameters of the incident wave, so that the effect of difference in form is eliminated. As a result, an analytical formula suitable for applications, in particular, in problems related to tsunamis, has been proposed for the height of run-up of a solitary wave on a beach.     

Interaction of ocean waves with floating and submerged horizontal flexible structures in three-dimensions

A three-dimensional general mathematical hydroelastic model dealing with the problem of wave interaction with a floating and a submerged flexible structure is developed based on small amplitude wave theory and linear structural response. The horizontal floating and submerged flexible structures are modelled with a thin plate theory. The linearized long wave equations based on shallow water approximations are derived and results are compared. Three-dimensional Green’s functions are derived using fundamental source potentials in water of finite and infinite depths. The expansion formulae associated with orthogonal mode-coupling relations are derived based on the application of Fourier transform in finite and infinite depths in case of finite width in three-dimensions. The usefulness of the expansion formula is demonstrated by analysing a physical problem of surface gravity wave interaction with a moored finite floating elastic plate in the presence of a finite submerged flexible membrane in three-dimensions. The numerical accuracy of the method is demonstrated by computing the complex values of reflected wave amplitudes for different modes of oscillation and mooring stiffness. Further, the effect of compressive force and modes of oscillations on a free oscillation hydroelastic waves in a closed channel of finite width and length for floating and submerged elastic plate system is analysed.     

Modelling of run up of steep non-breaking waves

A Navier–Stokes solver is used to examine steep waves as they run up a steep beach (10.54°). The volume of fluid method (VOF) is used to model the free surface. Comparison with experimental results shows reasonable overall agreement in the prediction of the free-surface, velocities and accelerations within the flow. A spurious feature at the free-surface was found which does reduce the quality of the results. For a steep wave we see the transition from a steep wave front to a smooth run-up tongue at the beach that is in qualitative agreement with experiment.     

Nonlinear Effect of Wave Propagation in Shallow Water

—In this paper,a nonlinear model is presented to describe wave transformation in shallow wat-er with the zero-vorticity equation of wave-number vector and energy conservation equation.Thenonlinear effect due to an empirical dispersion relation(by Hedges)is compared with that of Dalrymple＇sdispersion relation.The model is tested against the laboratory measurements for the case of a submergedelliptical shoal on a slope beach,where both refraction and diffraction are significant.The computation re-sults,compared with those obtained through linear dispersion relation.show that the nonlinear effect ofwave transformation in shallow water is important.And the empirical dispersion relation is suitable for re-searching the nonlinearity of wave in shallow water.     

Field investigation on the results of non-linear shallow water equations in the Swash Zone

For the study of the cross-shore wave-induced hydrodynamics in the swash zone, a numerical model is developed based on the one-dimensional non-linear shallow water (NSW) equations for prediction of hydrodynamic parameters in the swash zone. In order to evaluate the accuracy of the outputs of the numerical model, the model's predictions in terms of water surface elevations and cross-shore velocities, are compared to field data from full-scale experiments conducted on three sites with different beach slope; mild and steep, several bed particle sizes and under various incident wave conditions. The quantitative and qualitative comparison of the results of the numerical model and the full-scale data reveals that the model can generally predict many aspects of the flow in the surf and swash zone on both types of beach. The accuracy is adequate for application in a sediment transport study. Considering the time-history and probability distribution of water surface elevation, the model is generally more accurate on steep beaches than on the mild beach. The model can adequately simulate the dominant frequency across the beach and saturation of higher frequencies on both mild and steep beaches for various incident wave energy characteristics. With regard to the horizontal (cross-shore) velocity, the sawtooth shape of time-history and negative acceleration of water are well predicted by the model for both mild and steep beaches. Due to the uncertainties in maximum and minimum values of velocity data, clear judgement about the accuracy of the numerical model in this matter was not possible. However, the comparison of the minimum velocities (offshore direction) revealed that the application of friction factors below the range which is suggested by literature best match the data.     

显式非线性弥散关系在浅水波变形计算中的应用

本文参照Ｚｈａｏ和Ａｎａｓｔａｓｉｏｕ的方法,导出了逼近Ｂｏｏｉｊ的非线性弥散关系的近似显式表达式,该式给出的结果与Ｂｏｏｉｊ的非线性弥散关系相当吻合。用中文显式非线性弥散关系,结合会弱非线性效应的缓坡方程,构成含非线性影响项缓坡方程的一个求解浅水波变形问题的方程组。用实验数据对本文模型进行验证,结果表明,显式非线性弥散关系在求解浅水波变形问题时,给出了更符合实验数据的结果。     

海滩体积时空变化和波浪场的奇异谱分析

海滩响应波浪动力作用的变化是海岸带陆海相互作用研究的重要内容.利用奇异谱分解方法对南湾海滩连续实测剖面的体积变化值与南湾逐日波高变化值的相关关系进行了探讨,以诊断分析波浪在海滩变化过程中的内在作用机制,结果表明:(1)海滩体积的时空变化与波浪作用力有明显的相关关系;(2)海滩响应波浪动力作用变化最明显的地方是在中间岸段,在直线岸段次之,在遮蔽岸段最弱;(3)对海滩变化有较强作用的波浪场分别位于近岸及其与上下岬角连线的邻近区域.     

考虑边界波浪方向的缓坡方程自适应求解模型

缓坡方程是描述近岸波浪运动较好的数学模型之一。在发展的自适应有限元求解缓坡方程的基础上,采用迭代求解的方法,确定波浪相对于边界的入射方向,从而对边界条件进行改进,建立了求解缓坡方程的数值计算模型。典型算例表明,考虑波浪相对于边界的入射角度后,模型可以更好地模拟吸收波浪边界,同时对多向波对双突堤的绕射进行了模拟研究,与试验结果比较表明,所建立的数值计算模型能够适用于多向不规则波传播过程的模拟研究。     

浅水浮式生产储油系统(FPSO)二阶定常力数值仿真研究

应用三维线性势流理论和Ir J A Pinkster的近场分析方法,对超大型浮式生产储油系统(32万吨FPSO)的二阶波浪定常力进行了数值仿真分析,研究浅水情况下,不同水深对FPSO受到二阶波浪定常力的影响,对浅水油田中FPSO的设计应用有一定的实用意义。     

On low-frequency waves in the surf and swash

Low-frequency waves in the surf and swash zones on various beach slopes are discussed using numerical simulations. Simulated surface elevations of both primary waves and low-frequency waves across the surf zone were first compared with experimental data and good agreement found. Low-frequency wave characteristics are then discussed in terms of their physical nature and their relationship to the primary wave field on a series of sea bottom slopes. Unlike primary waves, low-frequency wave energy increases towards the shoreline. Low-frequency waves in the surf and swash are a function of incident waves and the sea bottom slope and hence the saturation level of the surf zone. Wave energy on a gently sloping beach is dominated by low-frequency waves while primary waves play a significant role on a steep beach. Low-frequency wave radiation from the surf zone on a given beach depends on primary wave frequency and beach slope. However, a very poor correlation was found between surf similarity parameter and low-frequency wave radiation.     

粤东后江湾冲流带滩面高频振动及动力作用分析

利用涌浪影响下短时段内的冲流带滩面高频高程数据和碎波带波流资料,在奇异谱分析(SSA)的基础上,对比研究了不同形态滩面的冲淤变化趋势、趋势分布形状、冲淤变化周期和冲淤变化强度,以及同一条剖面不同桩点间各因素间的变化关系;用交叉谱方法探索了每分钟滩面高频冲淤变化与碎波带长重力波间的作用关系。分析结果表明,滩角韵律地形引起的冲流分流作用促进了滩脊向滩谷的泥沙转运,冲流带滩面存在明显的长重力波频段的周期性冲淤振动,滩面冲淤振动强度由滩面下部向上部递减,碎波带长重力波对滩面高频冲淤变化起重要作用。     

Shoaling Surface Gravity Waves Cause a Force and a Torque on the Bottom

Freely propagating surface gravity waves are observed to slow down and to stop at a beach when the bottom has a relatively gentle upward slope toward the shore and the frequency range of the waves covers the most energetic wind waves (sea and swell). Essentially no wave reflection can be seen and the measured reflected energy is very small compared to that transmitted shoreward. One consequence of this is that the flux of the wave’s linear momentum decreases in the direction of wave propagation, which is equivalent to a time rate of change of the momentum. It takes a force to cause the time rate of change of the momentum. Therefore, the bottom exerts a force on the waves in order to decrease the momentum flux. By Newton’s third law (action equals reaction) the waves then impart an equal but opposite force to the bottom. In shallow (but finite) water depths the wave force per unit bottom area is calculated, for normal angle of incidence to the beach, to be directly proportional to the square of the wave amplitude and to the bottom slope and inversely proportional to the mean depth; it is independent of the wave frequency. Constants of proportionality are: 1/4, the fluid density and the acceleration of gravity. Swell attenuation near coasts and some characteristics of sand movement in the near-shore region are not inconsistent with the algebraic structure of the wave force formula. Since the force has a depth variation which is significantly faster than that of the dimensions of the particle orbits in the vertical direction, the bottom induces a torque on the fluid particles that decreases the angular momentum flux of the waves. By an extension of Newton’s third law, the waves also exert an equal but opposite torque on the bottom. And because the bottom force on the waves exists over a horizontal distance, it does work on the waves and decreases their energy flux. Thus, theoretically, the fluxes of energy, angular and linear momentum are not conserved for shoaling surface gravity waves. Mass flux, associated with the Stokes drift, is assumed to be conserved, and the wave frequency is constant for a steady medium.     

Data-based yearly forecasting of beach volumes along the Dutch North Sea coast

Beach and nearshore levels have been measured yearly along the entire Dutch North Sea coast since the mid 1960s (the ‘Jarkus’ data set). This data set has been processed to create separate time series of beach volumes at longshore intervals of about 250 m, giving over 2000 time series in total. These time series typically show a high annual variability with weak long-term trends. The present Dutch national coastal management strategy involves making year-ahead forecasts of beach volumes by extrapolating a linear least squares trend through the previous ten years' data separately for each longshore location. In this paper, these forecasts are shown to be worse than the trivial forecast in which the most recently measured beach volume persists unchanged into the future, with a mean square error (MSE) about 13.5% worse (equivalent to a root mean square error (RMSE) 6.5% worse). Improvements to these forecasts are sought by testing six different univariate forecasting methods. The two best methods improve on the persistence of the most recently measured beach volume by about 15% MSE (8% RMSE), and on the presently used linear least squares trend method by about 25% MSE (13.5% RMSE). Further comparisons are made between the forecasting methods to investigate several factors. These include varying the amount of fitting data for the forecasting methods, smoothing of the fitting data, different methods for interpolating gaps in the data, the longshore aggregation of data, making forecasts for coastal profiles with and without nourishments, and making forecasts up to five years ahead. These forecasting methods are designed as a coastal management tool to provide yearly forecasts quickly and routinely for the whole Dutch North Sea coast.     

Beach memory and ensemble prediction of shoreline evolution near a groyne

In this paper we address the question of estimating the average position of a beach and its inherent variability about this mean. It is demonstrated how, even in a much simplified situation, the ensemble average of beach plan shape involves cross-correlation of the beach position and wave conditions. This renders the governing equations inimical to analytical treatment. A new analytical expression for the mean beach plan shape and its variation are derived for the case of a single groyne exposed to waves varying in direction only. This demonstrates that ‘beach memory’ is directly related to the autocorrelation of wave direction. For more general conditions a semi-analytical expression for the ensemble average of the shoreline position is derived. This solution is estimated with site specific wave conditions using Monte Carlo simulations. The characteristics of the solution are investigated and it is demonstrated that, for this case at least, the terms involving the wave direction are virtually uncorrelated with the terms that do not. It is concluded that, in an ensemble sense, the morphodynamic impact of wave direction is decoupled from that due to wave height and period.     

Numerical investigation of secondary load cycle and ringing response of a vertical cylinder

A numerical wave tank is established based on two-phase FVM model and VOF method and verified with the physical experiment in Grue et al. (1994). Focusing waves with different wave steepness passing a vertical cylinder are investigated by Numerical simulations. The phenomenon called ‘secondary load cycle’ which may lead to ringing response, is observed and discussed. The presence of secondary load cycle could be related to Froude Number (Fr). The possible transition region of the present and absent secondary load cycle is Fr = 0.4. Sub and super harmonic wave components appear in the propagation of waves, second-order wave theory could give a good prediction. Morison equation with linear wave theory could predict well the wave forces of vertical cylinder with small steepness without the secondary load cycle, but cannot capture the crests/troughs of the wave forces with the secondary load cycle. Crest improvements are achieved by second-order wave theory. A spectral analysis based on wavelet transform is applied to wave loads. The frequency of the secondary load may be up to 13 times the wave frequency, which may cause the ringing response expanding to a higher frequency range. Strong ringing response occurs in steep wave, it could be extended up to 15 times wave trough-to-trough frequency due to the secondary load cycle. The damping has slightly influence on the peak of resonance response, but it will lead to faster decay of subsequent response, if the damping ratio is large.     

Morphodynamics of a bar-trough surf zone

A field study was made of the distinguishing morphodynamic processes operating in a surf zone which perennially exhibits accentuated bar-trough topography (the “longshore-bar-trough” and “rhytmic-bar-and-beach” states as described by Wright and Short, 1984). Characteristic features of the morphology include a shallow bar with a steep shoreward face, a deep trough, and a steep beach face. This morphology, which is favored by moderate breaker heights and small tidal ranges, strongly controls the coupled suite of hydrodynamic processes. In contrast to fully dissipative surf zones, the bar-trough surf zone is not at all saturated and oscillations at incident wave frequency remain dominant from the break point to the subaerial beach. The degree of incident wave groupiness does not change appreciably across the surf zone. Infragravity standing waves which, in dissipative surf zones, dominate the inshore energy, remain energetically secondary and occur at higher frequencies in the bar trough surf zone. Analyses of the field data combined with numerical simulations of leaky mode and edge wave nodal—antinodal positions over observed surf-zone profiles, indicate that the frequencies which prevail are favored by the resonant condition of antinodes over the bar and nodes in the trough. Standing waves which would have nodes over the bar are suppressed. Sediment resuspension in the surf zone appears to be largely attributable to the incident waves which are the main source of bed shear stress. In addition, the extra near-bottom eddy viscosity provided by the reformed, non-breaking waves traversing the trough significantly affects the vertical velocity profile of the longshore current. Whereas the bar is highly mobile in terms of onshore—offshore migration rates, the beach face and inner regions of the trough are remarkably stable over time.     

On the numerical modeling of tsunami wave generation and propagation

Abstract

In this article three main stages of tsunami wave evolution are investigated. At first, the development of disturbances from a given patched elevation of the bottom surface in an incompressible nonviscous fluid of the uniform depth is considered. Then, a tsunami wave diffraction by underwater bottom elevation or cavity is investigated. In this case the shallow water equations are already used, and it is supposed that a cylindrical wave is spread from patched water elevation over the epicentrum. Last, the tsunami propagation and transformation in a shallow water region and its run‐up on a beach are investigated on the basis of the improved shallow water theory, taking into consideration the nonlinear and dispersive terms of higher order. The proposed theory is tested in a problem of collisions of two solutions. Solutions of the first and the second problems are obtained by the method of integral Laplace's transformation with following numerical inversion of transformations. A finite difference method for a solution of the last problem is used.     

Formation of Equilibrium Beach Profile of the Abandoned Yellow River Delta Coast in North Jiangsu

The abandoned Yellow River Delta coast is a typical erodible silty and muddy coast in China. The paper analyses the marine dynamic characteristics and the mechanism of beach erosion of this area. Analysis and calculation show that in this sea area wave and tidal current action should be considered. Based on the above analysis, an equilibrium beach profile calculation model is developed, in which the wave-current interaction is considered while sediment supply and sediment re-deposition are neglected. The model consists of four parts: (1) calculation of wave parameters, (2) calculation of velocity due to wave-current interaction at different water depth, (3) calculation of friction velocity and shear stress at different water depths, and (4) calculation of the amount of sediment erosion, erosion intensity and variation of beach profile. Calculated results are in good agreement with observed data. Finally, the evolution tendency is discussed and the equilibrium beach profile of this coast is calculated. B     

SWAN predictions of waves observed in shallow water onshore of complex bathymetry

SWAN model predictions, initialized with directional wave buoy observations in 550-m water depth offshore of a steep, submarine canyon, are compared with wave observations in 5.0-, 2.5-, and 1.0-m water depths. Although the model assumptions include small bottom slopes, the alongshore variations of the nearshore wave field caused by refraction over the steep canyon are predicted well over the 50 days of observations. For example, in 2.5-m water depth, the observed and predicted wave heights vary by up to a factor of 4 over about 1000 m alongshore, and wave directions vary by up to about 10°, sometimes changing from south to north of shore normal. Root-mean-square errors of the predicted wave heights, mean directions, periods, and radiation stresses (less than 0.13 m, 5°, 1 s, and 0.05 m³/s² respectively) are similar near and far from the canyon. Squared correlations between the observed and predicted wave heights usually are greater than 0.8 in all water depths. However, the correlations for mean directions and radiation stresses decrease with decreasing water depth as waves refract and become normally incident. Although mean wave properties observed in shallow water are predicted accurately, nonlinear energy transfers from near-resonant triads are not modeled well, and the observed and predicted wave energy spectra can differ significantly at frequencies greater than the spectral peak, especially for narrow-band swell.